Question

Differentiate cot^1 (sqrt(1+x^2) + x) with respect to x?

Answer 1

is it \[\cot (x) \sqrt{1+x ^{2}}+x \] ?

Answer 2

Sadly, no :P it's cot inverse.. Or u can say arc cot

Answer 3

Ohhh i got it.. That's a nice long product rule, chain rule, with trig identity all tied up into one huh?

Answer 4

And there is no x after cot^-1...

Answer 5

\[\cot^{-1}(\sqrt{1+x ^{2}}+x) \]

Answer 6

that right?

Answer 7

Exactly :)

Answer 8

Give me a minute to work through it myself. That's a good one. :-p

Answer 9

Ok man, cheers :)

Answer 10

\[-((x/\sqrt{x ^{2}+1})+1)/((\sqrt{x ^{2}+1}+x)^{2}+1)\]

Answer 11

Check that out...Just have to work through it step by step...Hopefully someone else checks that too to make sure I didn't miss something.

Answer 12

Hmm... So can u explain the steps involved, as in what exactly did u do?

Answer 13

Okay...Start with the derivative of the inside: \[\sqrt{1+x ^{2}}+x\]

Answer 14

Can you do that?

Answer 15

2x/2sqrt(1+x^2). +. 1. ?

Answer 16

Think about it as (1+x^2)^(1/2) + x

Answer 17

Oh

Answer 18

you will get x/sqrt(x^x +1) +1 Using the chain rule

Answer 19

Yeah that's what I've written...

Answer 20

Okay..and now do you know what the derivative of cot^-1(x)

Answer 21

-1/1+x^2?

Answer 22

and it if were a u-substitution it would be -1/1+u^2..with u=inside of the arccot...

Answer 23

So you end up with (1*x/sqrt(x^x +1) +1) / (u^2 +1) then plug in your u, which was sqrt(x^2 +1) + x

Answer 24

see it come together?

Answer 25

Got it @Dio ? I'm about to run.

Answer 26

Is it x^x or x^2?

Answer 27

x^2 sorry. the way I typed it the first time. lol

Answer 28

Trying to get a hang of it. Thanks!

Answer 29

No problem. That was a tough and tedious one that you just had to watch each piece one by one. You will get it....the more you do, the easier it is to see. :-) good luck.

Answer 30

Thanks a lot man, u rock :D

Answer 31

Glad I could help!

Answer 32

Oh wait.

Answer 33

It's waaaaay too easy. Don't start differentiating from the beginning. Use some properties of inverse trigonometry to get pi/2 - arccotx, and then differentiate.
Thanks for the help anyway, much appreciated.

Answer 34

Substitution - let x = cot a.